High flexible structure nonlinear aerodynamic damping estimation method, system and storage medium based on sage-husa algorithm and square root cubature kalman filter

Abstract

The application discloses a high-flexible-structure nonlinear aerodynamic damping estimation method based on a Sage-Husa algorithm and square root cubature Kalman filtering, which can not only effectively identify aerodynamic damping with nonlinear characteristics from free vibration data, forced vibration data, steady vibration data and random vibration data of high-flexible structures such as high-rise buildings and fan towers under transverse wind excitation, but also can automatically adjust process noise according to a current state, accurately predict parameters of interest of a high-dimensional system, and does not need to have enough knowledge of noise levels, so that the high-flexible-structure aerodynamic damping can be more conveniently and accurately calculated in actual engineering to provide a theoretical basis for evaluating structural responses. The application further discloses a high-flexible-structure nonlinear aerodynamic damping estimation system based on the Sage-Husa algorithm and the square root cubature Kalman filtering and a storage medium.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aerodynamic analysis, and particularly relates to a high-flex structure nonlinear aerodynamic damping estimation method, system and storage medium based on a Sage-Husa algorithm and a square root cubage Kalman filter. BACKGROUND

[0002] Aerodynamic damping refers to the energy dissipation effect of fluid flowing through a blunt body on structural vibration, and is an important aerodynamic force parameter affecting structural vibration. Due to the influence of multiple factors such as structural shape, flow field characteristics and amplitude dependence on the generation mechanism, it is very difficult to theoretically analyze and study aerodynamic damping. For high-flex structures (such as high-rise buildings, fan tower drums, etc.), near the lock-in range of vortex-induced vibration, aerodynamic damping usually shows a negative value and has significant amplitude dependence. In order to accurately evaluate the response of such structures under the action of crosswind in engineering practice, the key lies in accurately modeling the aerodynamic damping. An accurate aerodynamic damping model can not only improve the accuracy and reliability of the evaluation results, but also provide an effective basis for structural wind resistance design, thereby enhancing the safety and durability of buildings.

[0003] In recent years, the identification method of aerodynamic damping has made significant progress, mainly focusing on experiments, numerical simulation and data-driven methods, etc. In the experimental method, the free vibration test effectively extracts the aerodynamic damping through wind tunnel experiments combined with Hilbert transform (HT) and other technologies, and especially shows good results in capturing the nonlinear aerodynamic damping characteristics. While the forced vibration test uses random decrement technology (RDT) to obtain linear damping, but has limitations in identifying nonlinear damping. At the same time, the application of computational fluid dynamics (CFD) technology also provides a new idea for the extraction of nonlinear aerodynamic damping, which simulates the complex process of airflow and structure interaction to obtain more accurate aerodynamic damping values.

[0004] In terms of statistical and data-driven methods, techniques such as least squares and neural networks have been applied to damping identification, handling complex nonlinear characteristics and improving identification accuracy. In addition, nonlinear Kalman filter (NKF) and unscented Kalman filter (UKF) have been used for real-time identification of aerodynamic damping, effectively handling noise and nonlinear characteristics in dynamic systems. In some studies, an iterative unscented Kalman filter (UKF) is further used to extract the aerodynamic damping ratio from free vibration data, thereby significantly enhancing the robustness of the identification process when the initial conditions change. This method optimizes the iterative process, making the system more adaptable to noise and parameter changes. However, there is still an urgent need for a comprehensive and in-depth understanding of the vibration system in the current research on aerodynamic damping identification. Such in-depth analysis is crucial for accurately specifying the characteristics of the process noise, as the nature of the noise directly affects the reliability of the estimation results. Only with a thorough understanding of the dynamic characteristics of the vibration system and the source of the noise can the effectiveness of the model and the accuracy of the results be ensured.

[0005] Compared with other nonlinear Kalman filter methods, the square root cubature Kalman filter (SRCKF) exhibits higher accuracy and efficiency in handling high-dimensional nonlinear systems. However, obtaining ideal results requires a proper understanding of the characteristics of the process noise. In practical applications, noise characteristics are often unknown and dynamically changing, which can significantly reduce the estimation accuracy and even cause the system to diverge if the noise model is inaccurate. SUMMARY

[0006] Therefore, the purpose of the present application is to provide a high-soft structure nonlinear aerodynamic damping estimation method, system and storage medium based on Sage-Husa algorithm and square root cubature Kalman filter, which can automatically adjust the process noise according to the current state and accurately predict the parameters of interest in high-dimensional systems without the need for sufficient knowledge of the noise level, providing a theoretical basis for more accurate calculation of the aerodynamic damping of flexible structures in practical engineering.

[0007] To achieve the above purpose, the present application provides the following technical solutions:

[0008] The present application first proposes a high-soft structure nonlinear aerodynamic damping estimation method based on Sage-Husa algorithm and square root cubature Kalman filter, characterized by the following steps:

[0009] Step one: based on the high-soft structure cross-wind response equation and the harmonic balance technique, the relationship between nonlinear aerodynamic damping and structural vibration amplitude is obtained;

[0010] Step two: define the measurement equation by vibration displacement, velocity and acceleration; define the state space equation by considering the displacement, velocity, frequency, unknown structural aerodynamic damping parameters and external load of the high flexible structure;

[0011] Step three: time update: firstly define the initial state estimation value and the initial state covariance matrix P0, generate the cubature points X k / k by using the spherical radial cubature rule; then propagate the cubature points by the state transition function f(·) to predict the state mean value and the square root of state covariance S k+1 / k ;

[0012] Step four: measurement update: firstly recalculate the cubature points by using and S k+1 / k , then propagate the cubature points by the measurement function h(·) to obtain the predicted mean value the square root of innovation covariance S zz,k+1 / k and the cross covariance P xz,k+1 / k ;

[0013] Step five: state update: calculate the Kalman gain K k+1 , update the state matrix and the square root of covariance S k+1 .

[0014] Step six: noise update: evaluate whether the state space equation parameters have stabilized: if yes, it indicates that the estimation has reached the best performance, and the process noise remains unchanged; if not, it indicates that the state space equation parameters continue to fluctuate, and the Sage-Husa estimator is used to update the process noise Q in real time until the state space equation parameters reach the stable state;

[0015] Step seven: repeat steps three to six for each sample point until all sample points are processed to obtain stable nonlinear aerodynamic damping parameters;

[0016] Step eight: calculate the nonlinear aerodynamic damping by using the obtained stable nonlinear aerodynamic damping parameters.

[0017] Further, in the step one, the relationship between the nonlinear aerodynamic damping and the structural vibration amplitude is:

[0018]

[0019] wherein: ξ a is the nonlinear aerodynamic damping; ρ is the air density; m is the structural width; m s represents the effective structural mass per unit height; y max is the structural vibration amplitude; B1, B2, B3, B4 and B5 are nonlinear aerodynamic damping parameters, and are all constants.

[0020] Further, the measurement equation is defined as:

[0021]

[0022] where: y, and denote displacement, velocity, and acceleration;

[0023] The state space equation is:

[0024]

[0025] where: B1', B2', B3', B4', and B5 are structural aerodynamic damping parameters; S is a defined zero-mean Gaussian white noise external load; ω s = 2πf s denotes the modal circular frequency, f s is the frequency; and: B1, B2, B3, B4, and B5 are nonlinear aerodynamic damping parameters, and are all constants.

[0026] Further, in step three, the time update method step is:

[0027] 31) define the initial state estimate value and the initial state covariance matrix P0 are defined as:

[0028]

[0029] where: X0 denotes the initial state space equation; E(·) denotes taking the expectation; COV(·) denotes taking the covariance.

[0030] The state space vector and the measurement vector are obtained:

[0031] X k = f(X k-1 ) + w k

[0032] Z k = h(X k ) + v k

[0033] where: X k ∈ R n and Z k ∈ R m are the state vector and the measurement vector at time step k, respectively; f(·) and h(·) denote the nonlinear state transition function and the nonlinear measurement function; k is the current time step; w k ∈ R n and vk ∈R m are uncorrelated zero-mean Gaussian white noises with covariance matrices and Q k is the process noise covariance; R k is the measurement noise covariance;

[0034] 32) Generate volume points X k / k :

[0035]

[0036] where S k / k is the orthogonal-triangular decomposition of P k / k , P k / k is the state covariance matrix; ξ is the set of volume points; e i is the i-th column of the identity matrix I n .

[0037] 33) Propagate volume points through the state transition function f(·):

[0038]

[0039] where: is the new volume point at time step k+1 obtained by propagating through the state transition function; X i,k / k is the volume point at time step k.

[0040] 34) Predict the state mean and the square root of the state covariance S k+1 / k :

[0041]

[0042] where:

[0043] S Q = chol(Q)

[0044]

[0045] where chol denotes the Cholesky decomposition function; Tria(·) denotes the QR decomposition of the matrix; S Q is the Cholesky decomposition of Q; Q is the process noise; is the weight matrix.

[0046] Further, in the fourth step, the method step of measurement update is:

[0047] 41) Use the state mean and the square root of the state covariance Sk+1 / k Recalculate volume points:

[0048]

[0049] 42) Propagate volume points by measurement function h(·):

[0050] Z i,k+1 / k = h(X i,k+1 / k ), i = 1, 2,..., 2n

[0051] 43) Predict mean from transformed volume points New innovation covariance square root S zz,k+1 / k and cross covariance P xz,k+1 / k

[0052]

[0053] where: S R is the Cholesky decomposition of R, R is the measurement noise; η k+1 / k is the measurement weight matrix; Z k+1 / k is the new volume points obtained by propagating through the measurement function.

[0054] Further, in the step five, the Kalman gain K k+1 is:

[0055]

[0056] The state matrix is updated as:

[0057]

[0058] The square root of state covariance S k+1 is updated as:

[0059] S k+1 = Tria([χ k+1 / k - K k+1 η k+1 / k , K k+1 S R ]).

[0060] Further, in the step six, the method for updating the noise is:

[0061] Update the process noise Q using Sage-Husa estimator:

[0062]

[0063] where:

[0064]

[0065] Where: b is the forgetting factor; α1 is the introduced empirical proportionality factor; e k+1 For the new information sequence;

[0066] After the noise Q in the Sage-Husa estimator iteration process reaches a preset number of k iterations, the data accumulated during these k iterations are used to evaluate whether the parameters of the state-space equation have stabilized.

[0067] If in the current j-th iteration, j > k, and the ratio of the standard deviation to the mean of the parameters satisfies:

[0068]

[0069] Where: std j The mean represents the standard deviation of the parameters in the j-th iteration. j std(X) represents the average parameter value in the j-th iteration. j-k:j ) represents the standard deviation of the parameters from the current j-th iteration to the previous jk iterations; mean(X) j-k:j ) represents the average value of the parameters from the current j-th iteration to the previous jk iterations; δ is the preset threshold.

[0070] The state-space equation parameters are now stable, and the process noise remains constant.

[0071] Q j+1 =Q j

[0072] Otherwise, it indicates that the state-space equation parameters are unstable, and the process noise Q is updated as follows:

[0073]

[0074] Where α2 is the introduced empirical scaling factor.

[0075] This invention also proposes a highly flexible nonlinear aerodynamic damping estimation system based on the Sage-Husa algorithm and square root ductile Kalman filter, including a memory, a processor, and a computer program stored in the memory and executable on the processor. The processor executes the computer program and implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter as described above.

[0076] The present invention also proposes a storage medium storing a computer program, which, when executed by a processor, implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root commutative Kalman filtering as described above.

[0077] The beneficial effects of this invention are as follows:

[0078] The key to improving the application effect of SRCKF is to build a noise model that is more suitable for actual conditions by deeply studying the dynamic characteristics of noise characteristics. The Sage-Husa estimator is used to estimate the noise statistics in real time, and by introducing the Sage-Husa estimator, the NKF can estimate the process noise covariance matrix in the nonlinear system in real time. The Sage-Husa estimator is combined with the SRCKF, which helps to deeply explore the accuracy and system stability of the filter estimation under dynamic noise conditions. Therefore, the integration of SRCKF and Sage-Husa estimator has important significance, which not only helps to improve the performance of the filter in the unknown noise environment, but also provides new theoretical support for real-time accurate estimation of high-dimensional nonlinear systems. That is, the high flexible structure nonlinear aerodynamic damping estimation method based on Sage-Husa algorithm and square root cubature Kalman filter can not only effectively identify the aerodynamic damping with nonlinear characteristics from the free vibration data, forced vibration data, steady vibration data and random vibration data of high flexible structures such as high-rise buildings and fan towers under transverse wind excitation, but also can automatically adjust the process noise according to the current state and accurately predict the parameters of interest of high-dimensional systems without sufficient understanding of the noise level, which can provide a theoretical basis for more convenient and accurate calculation of aerodynamic damping of high flexible structures in practical engineering to evaluate the structural response. BRIEF DESCRIPTION OF DRAWINGS

[0079] In order to make the purpose, technical scheme and beneficial effects of the present application clearer, the present application provides the following drawings for illustration:

[0080] Figure 1 It is a schematic diagram of a high flexible circular tower structure;

[0081] Figure 2 It is a flowchart of the high flexible structure nonlinear aerodynamic damping estimation method based on Sage-Husa algorithm and square root cubature Kalman filter (SSRCKF) of the present application;

[0082] Figure 3 It is a structural damping ξ s =0.5% and the random transverse wind response data of the reduced wind speed U / fB=11.7;

[0083] Figure 4 It is a comparison diagram of the measured displacement and the SSRCKF estimated displacement;

[0084] Figure 5 It is a comparison diagram of the measured velocity and the SSRCKF estimated velocity;

[0085] Figure 6 It is a comparison diagram of the parameter value of the nonlinear aerodynamic damping estimated by SSRCKF and the target value;

[0086] Figure 7 This is a comparison between the estimated aerodynamic damping values ​​at different times, obtained using SSRCKF, and the target values.

[0087] Figure 8 For structural damping ξ s Different types of response data with a wind speed of 0.5% and a reduced wind speed U / fB of 11.7;

[0088] Figure 9 This is a comparison of the aerodynamic damping extracted by SSRCKF from different types of data with the target value. Detailed Implementation

[0089] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0090] In this embodiment, the proposed nonlinear aerodynamic damping estimation method for highly flexible structures based on the Sage-Husa algorithm and square root ductile Kalman filter (SSRCKF) is applied to calculate the nonlinear aerodynamic damping of a highly flexible circular tower structure. The effectiveness and accuracy of the estimated nonlinear aerodynamic damping are evaluated by comparing it with the true value. Specifically, the circular tower structure has a height H of 200m, an aspect ratio H / B = 13.3, and a mass parameter m. s / ρB 2 =172, modal angular frequency ω s =1.26, the structural damping ratio has ξ s =0.5%, 1%, and 2%, assuming the fundamental mode shape has a linear relationship along the structural height, i.e., φ(y) = y / H, such as Figure 1 As shown.

[0091] like Figure 2 As shown, this embodiment describes a highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root commensurate Kalman filter (SSRCKF), which includes the following steps.

[0092] Step 1: Based on the crosswind response equation of highly flexible structures and harmonic balance technology, the relationship between nonlinear aerodynamic damping and structural vibration amplitude is obtained.

[0093] Specifically, the relationship between aerodynamic damping and structural vibration amplitude is derived using the structural crosswind response equation and harmonic balance techniques. The crosswind response equation of the flexible structure based on the first modal response is expressed as:

[0094]

[0095] Where: y1 represents the generalized displacement (linear or torsional displacement) of the structure; For speed; is the acceleration; ω s = 2πf s denotes the modal circular frequency, f s is the frequency; ξ s denotes the structural damping ratio; ξ a denotes the aerodynamic damping ratio; Q(t) denotes the generalized crosswind force; M s denotes the generalized mass.

[0096] A high-frequency force balance model test is performed, and when the generalized crosswind force Q(t) is estimated from the base bending moment M(t), the relationship is:

[0097]

[0098] wherein: η b is an empirical modal shape correction factor, and for linear modal shapes η b = 1 is generally set; H denotes the height of the structure; ρ is the air density; B is the width of the structure; m s denotes the effective structural mass per unit height; U is the wind speed at the top of the structure; η is a dimensionless parameter of the modal shape, and for linear modal shapes η = 3; C Mb (t) denotes the base moment coefficient.

[0099] The aerodynamic damping ξ a is denoted as:

[0100]

[0101] wherein: B1, B2, B3, B4 and B5 are nonlinear aerodynamic damping parameters, and are all constants; y = y1(t) / B denotes the normalized dimensionless displacement. A low-order or high-order polynomial model can be used according to requirements.

[0102] Using the harmonic balance technique y = y max sin(ωt), the relationship between the nonlinear aerodynamic damping and the structural vibration amplitude is:

[0103]

[0104] wherein: ξ a is the nonlinear aerodynamic damping; ρ is the air density; B is the width of the structure; m s denotes the effective structural mass per unit height; y max is the structural vibration amplitude; B1, B2, B3, B4 and B5 are nonlinear aerodynamic damping parameters, and are all constants.

[0105] In this embodiment, a 3-order polynomial model is used to estimate the nonlinear aerodynamic damping of the high-flexibility circular tower structure.

[0106] Nonlinear aerodynamic damping ξ represented by a 3rd order polynomial model a is:

[0107]

[0108] where B1, B2, B3 are parameters fitted by SSRCKF estimation.

[0109] Using harmonic balance technique y = y max sin(ωt), aerodynamic damping ξ a is related to vibration amplitude as:

[0110]

[0111] Step 2: Define measurement equation by vibration displacement, velocity and acceleration; consider displacement, velocity, frequency, unknown structural aerodynamic damping parameters and external load of high flexible structure, and define state space equation.

[0112] Consider structural displacement, velocity, frequency, unknown structural aerodynamic damping parameters and external load, and the state space equation is:

[0113]

[0114] where y represents structural displacement; is velocity; B'1, B'2, B'3, B'4 and B'5 are structural aerodynamic damping parameters; S is defined zero mean Gaussian white noise external load, and:

[0115]

[0116] The nonlinear state space equation can be solved by the fourth order Runge-Kutta method, and is represented as:

[0117]

[0118] According to the measured displacement or acceleration time history, the vibration velocity time history is obtained by derivation or integration, and then the SSRCKF method is used for post-processing.

[0119] Taking vibration displacement, velocity and acceleration as measurement variables can improve the estimation accuracy. Specifically, the measurement equation is defined as:

[0120]

[0121] where y, and represent displacement, velocity and acceleration.

[0122] Specifically, in this embodiment, after calculation using a 3rd order polynomial model, the measurement equation and the state space equation are represented as:

[0123] Considering structural displacement, velocity, frequency, unknown structural aerodynamic damping parameters and external load, the state space equation is:

[0124]

[0125] where y represents the displacement of the top of the circular tower structure, is the velocity, ω s is the modal circular frequency, and S is the defined zero-mean Gaussian white noise external load.

[0126] The nonlinear state space equation can be solved by the fourth-order Runge-Kutta method and is expressed as:

[0127]

[0128] In actual engineering, the measurement values that can be obtained for the built structure are generally only vibration accelerations accompanied by large noise; and in wind tunnel tests, vibration displacement or vibration acceleration can be directly measured, and the noise level can become smaller. In subsequent applications, the vibration velocity time history can be obtained by derivation or integration according to the displacement or acceleration time history obtained by measurement or experiment, and the influence of noise is processed, and then the SSRCKF method is used for post-processing.

[0129] Step three: time update: first define the initial state estimate and the initial state covariance matrix P0, and generate the volume points X k / k by using the spherical radial volume rule; then propagate the volume points by using the state transition function f(·) to predict the state mean and the square root S k+1 / k of the state covariance.

[0130] In this embodiment, the random transverse wind response data of the structural damping ξ s = 0.5% and the reduced wind speed U / fB = 11.7 are used to verify the accuracy of the SSRCKF method in identifying the nonlinear aerodynamic damping. The target value of the aerodynamic damping parameter under this condition is: Figure 3

[0131] [B1 B2 B3] = [-4.59 123.95 -484.39]

[0132] Specifically, in this embodiment, the method steps of time update are:

[0133] 31) Before time update, define the initial state estimate and the initial state covariance matrix P0: ​

[0134]

[0135] Where: X0 represents the initial state space equation; E(·) represents taking the expectation; COV(·) represents taking the covariance.

[0136] Using the SSRCKF technique, the initial state space vector can be set to any value without affecting the final estimation. In this embodiment, the initial value X0 is selected as [0 0.2 0 0 0 0 0]. T The initial covariance matrix P0 can be chosen arbitrarily, but its size will affect the convergence speed of the SSRCKF estimation. It is recommended to use a larger P0 value to accelerate convergence; P0 = 10 is suggested. 3 diag([1 1 1 1 1 11 1]). Additionally, the covariance Q of the process noise is defined. k and measurement noise covariance R k In nonlinear Kalman filtering, the choice of process noise Q is crucial for accurate estimation. However, the characteristics of process noise are often unknown, making accurate modeling difficult. Inappropriate process noise not only reduces computational accuracy but also causes filter divergence, which is particularly problematic in high-dimensional systems. In this embodiment, the Sage-Husa estimator can estimate and dynamically update the process noise in real time based on its statistical properties. Therefore, the process noise can take any value, as it automatically converges to a suitable value. In this embodiment, the initial process noise Q0 = 10 is selected. 3 diag([1 1 1 1 1 1 1 1]). The measurement noise can be set to a lower value, such as R0 = 10. - 3 diag([1 1 1]).

[0137] In a nonlinear discrete system, the state-space vector and measurement vector are obtained:

[0138] X k =f(X) k-1 )+w k

[0139] Z k =h(X) k )+v k

[0140] Where: X k ∈R n and Z k ∈R m Let f(·) and h(·) be the state vector and measurement vector at time step k, respectively; f(·) and h(·) represent the nonlinear state transition function and the nonlinear measurement function, respectively; k is the current time step; w k ∈R n and vk ∈R m These are the covariance matrices. and Uncorrelated zero-mean Gaussian white noise; Q k R represents the process noise covariance. k To measure the noise covariance.

[0141] 32) Generate the volume point X using the radial volume rule of a sphere. k / k For a state dimension n, the volume point is defined as:

[0142]

[0143] Wherein: S k / k It is P k / k orthogonal-triangular decomposition, P k / k Let ξ be the state covariance matrix; ξ be the set of volume points; e i It is the identity matrix I n The i-th column.

[0144] 33) Propagation of volume points through the state transition function f(·):

[0145]

[0146] in: X is the new volume point at time step k+1 obtained through propagation via the state transition function; i,k / k Let k be the volume point at time step k.

[0147] 34) Predicted state mean And the square root of the state covariance S k+1 / k :

[0148]

[0149] in:

[0150] S Q =chol(Q)

[0151]

[0152] Where: chole represents the Cholesky decomposition function; Tria(·) represents the QR decomposition of the matrix; S Q The Cholesky decomposition of Q; Q represents process noise; This is the weight matrix.

[0153] Step 4: Measurement Update: First use and S k+1 / kThe volume points are recalculated, and then propagated through the measurement function h(·) to obtain the predicted mean. The square root of the new covariance S zz,k+1 / k and cross-covariance P xz,k+1 / k .

[0154] Specifically, in this embodiment, the measurement update method steps are as follows:

[0155] 41) Use state mean The square root of the state covariance S k+1 / k Recalculate the volume points:

[0156]

[0157] 42) Propagate the volume point through the measurement function h(·):

[0158] Z i,k+1 / k =h(X) i,k+1 / k ), i = 1, 2, ..., 2n

[0159] 43) Predict the mean from the transformed volume points The square root of the new covariance S zz,k+1 / k and cross-covariance P xz,k+1 / k

[0160]

[0161] Wherein: S R Let R be the Cholesky decomposition of R, where R is the measurement noise; η k+1 / k For measuring the weight matrix; Z k+1 / k This is the new volume point obtained through propagation of the measurement function.

[0162] Step 5: State Update: Calculate Kalman Gain K k+1 Update the state matrix The square root of the covariance S k+1 .

[0163] Specifically, Kalman gain K k+1 for:

[0164]

[0165] State matrix Updated to:

[0166]

[0167] The square root of the state covariance S k+1 Updated to:

[0168] S k+1 =Tria([χk+1 / k -K k+1 η k+1 / k ,K k+1 S R ])

[0169] Step Six: Noise Update: Evaluate whether the state-space equation parameters have stabilized: if yes, it indicates that the estimation has reached optimal performance, keeping the process noise constant; if no, it indicates that the state-space equation parameters continue to fluctuate, and the Sage-Husa estimator is used to update the process noise Q in real time until the state-space equation parameters reach a stable state.

[0170] Specifically, the noise update method is as follows:

[0171] The process noise Q is updated using the Sage-Husa estimator:

[0172]

[0173] in:

[0174]

[0175] Where: b is the forgetting factor, which is taken as b = 0.95 in this embodiment; α1 is the introduced empirical scaling factor, which is taken as 1 in this embodiment based on numerical simulation and experimental verification; e k+1 This is a new information sequence.

[0176] After the noise Q in the Sage-Husa estimator iteration process reaches a preset number of k iterations, the data accumulated during these k iterations are used to evaluate whether the parameters of the state-space equation have stabilized.

[0177] If in the current j-th iteration, j > k, and the ratio of the standard deviation to the mean of the parameters satisfies:

[0178]

[0179] Where: std j The mean represents the standard deviation of the parameters in the j-th iteration. j std(X) represents the average parameter value in the j-th iteration. j-k:j ) represents the standard deviation of the parameters from the current j-th iteration to the previous jk iterations; mean(X) j-k:j ) represents the average parameter value from the current j-th iteration to the previous jk iterations; δ is the preset threshold.

[0180] The state-space equation parameters are now stable, and the process noise remains constant.

[0181] Q j+1 =Q j

[0182] Otherwise, it indicates that the state-space equation parameters are unstable, and the process noise Q is updated as follows:

[0183]

[0184] Wherein, α2 is an introduced empirical scaling factor. To accelerate parameter stabilization, the variation amplitude of process noise must be reduced. Therefore, in this embodiment, α2 is set to 10. -6 .

[0185] In this embodiment, the preset number of iterations k = 1000 is set during the noise update process, indicating that the parameter stability is evaluated after 40 seconds of free iteration. In the first 1000 free iterations of the data, SSRCKF not only updates the process noise Q in real time, but also accumulates data for subsequent evaluation of whether the parameters have stabilized.

[0186] After j (j>k=1000) iterations, the stability of the state space vector parameters was evaluated using the accumulated data. The stability criterion is that, in the current j-th iteration and the previous k=1000 iterations, the ratio of the standard deviation to the mean of the parameters satisfies:

[0187]

[0188] Where: std j The mean represents the standard deviation of the parameters in the j-th iteration. j std(X) represents the average parameter value in the j-th iteration. j-k:j ) represents the standard deviation of the parameters from the current j-th iteration to the previous j-1000 iterations; mean(X) j-k:j ) represents the average parameter value from the current j-th iteration to the previous j-1000 iterations; δ is a preset threshold. In this embodiment, δ = 5%.

[0189] Step 7: Repeat steps 3 to 6 for each sample point until all sample points have been processed and stable nonlinear aerodynamic damping parameters are obtained.

[0190] In this embodiment, the parameters of the nonlinear aerodynamic damping are estimated based on the first 600 seconds of displacement data. With a time step Δt = 0.04 s, the process iterates for 15,000 steps, following four steps: time update, measurement update, state update, and noise update, to obtain the parameters of the nonlinear aerodynamic damping estimated by SSRCKF. For example... Figure 4 The figure shown is a comparison between the measured displacement and the SSRCKF estimated displacement. Figure 5 The figure shown is a comparison between the measured velocity and the SSRCKF estimated velocity. It can be seen that the measured displacement and velocity are almost consistent with the SSRCKF estimated values. Figure 6The chart shows a comparison between the estimated values ​​of the three aerodynamic damping parameters B1, B2, and B3 using SSRCKF and their actual values. It can be observed that the parameter values ​​estimated by SSRCKF are almost identical to the actual target values ​​of the aerodynamic damping when the signal converges to a stationary state. These comparison charts demonstrate the accuracy of SSRCKF in response estimation.

[0191] Step 8: Calculate the nonlinear aerodynamic damping using the obtained stable nonlinear aerodynamic damping parameters.

[0192] Specifically, the obtained stable nonlinear aerodynamic damping parameters B1, B2, B3, B4, and B5 are substituted into the relationship between the aerodynamic damping ratio and the structural vibration amplitude:

[0193]

[0194] Nonlinear aerodynamic damping with amplitude dependence can be obtained by using amplitude and aerodynamic damping parameters.

[0195] In this embodiment, the three aerodynamic damping parameter values ​​B1, B2, and B3 obtained through the iterative process in step seven are substituted into the relationship between the aerodynamic damping ratio and the structural vibration amplitude:

[0196]

[0197] To evaluate the performance of the SSRCKF estimate, parameter estimates at five different times, t = 0s, 40s, 100s, 400s, and 500s, were used to calculate the nonlinear aerodynamic damping and compared with the target nonlinear aerodynamic damping value. Figure 7 As shown, it can be observed that when the parameter values ​​are stable, i.e., at t = 400s and 500s, the estimated aerodynamic damping values ​​are almost identical to the target values. To obtain more accurate aerodynamic damping estimates, the specific values ​​of parameters B1, B2, and B3 can be obtained by averaging the values ​​over the time interval after the estimated parameters have converged and stabilized.

[0198] SSRCKF can not only accurately estimate nonlinear aerodynamic damping from displacement data of random vibrations, but also obtain aerodynamic damping values ​​from other types of data, such as... Figure 8 The aerodynamic damping values ​​can also be obtained relatively accurately from the free vibration data, steady-state data, and forced vibration data shown (represented as Case 1, Case 2, and Case 3, respectively). Figure 9 The figure shows a comparison between the aerodynamic damping values ​​obtained using SSRCKF under these three different vibration conditions and the target values. It can be observed that SSRCKF can also accurately estimate nonlinear aerodynamic damping from vibration data of different types.

[0199] This embodiment uses a nonlinear aerodynamic damping estimation method for highly flexible structures based on the Sage-Husa algorithm and square root ductile Kalman filter (SSRCKF) to estimate the nonlinear aerodynamic damping of flexible structures. This method iterates through time updates, measurement updates, state updates, and noise updates to iterate through data points and predict actual values ​​with relatively high accuracy, thereby obtaining the relevant nonlinear aerodynamic damping parameters. A key feature of this method is its ability to automatically adjust process noise based on the current state and accurately predict parameters of interest in high-dimensional systems without requiring sufficient knowledge of the noise level, simplifying initial value setting. Furthermore, this method can obtain nonlinear aerodynamic damping parameters from different types of vibration data, providing a theoretical basis for more convenient and accurate calculation of aerodynamic damping in highly flexible structures such as towers in practical engineering to evaluate structural response. Finally, applying this method to estimate the nonlinear aerodynamic damping of a 200m high highly flexible circular tower structure under different vibration conditions reveals that SSRCKF can accurately predict the state values ​​of vibration data, and the estimated aerodynamic damping parameters converge to values ​​almost identical to the actual values. For different types of vibration data, such as free vibration data, steady-state data, and forced vibration data, the aerodynamic damping parameters can be estimated well, and nonlinear aerodynamic damping that is almost consistent with the target value can be obtained. It should be noted that the nonlinear aerodynamic damping estimation method for highly flexible structures based on the Sage-Husa algorithm and square root ductile Kalman filter (SSRCKF) in this embodiment is not only applicable to highly flexible circular tower structures, but also applicable to other highly flexible structures, such as wind turbine towers, high-rise buildings, and single-degree-of-freedom vibrations of long-span bridges.

[0200] This embodiment also proposes a highly flexible nonlinear aerodynamic damping estimation system based on the Sage-Husa algorithm and square root ductile Kalman filter, including a memory, a processor, and a computer program stored in the memory and run on the processor. The processor executes the computer program and implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter as described in this embodiment.

[0201] This embodiment also proposes a storage medium storing a computer program. When the computer program is executed by a processor, it implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root commutative Kalman filtering as described in this embodiment.

[0202] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filtering, characterized in that: Includes the following steps: Step 1: Based on the crosswind response equation of highly flexible structures and harmonic balance technology, the relationship between nonlinear aerodynamic damping and structural vibration amplitude is obtained; Step 2: Define the measurement equations using vibration displacement, velocity, and acceleration; considering the displacement, velocity, frequency, unknown aerodynamic damping parameters, and external loads of the highly flexible structure, define the state-space equations. Step 3: Time Update: First, define the initial state estimate. and the initial state covariance matrix The volume points are generated using the radial volume principle of a spherical surface. Then through the state transition function Propagation volume point, predicted state mean and the square root of the state covariance ; Step 4: Measurement Update: First use and Recalculate the volume point, then use the measurement function. Propagation volume points yield the predicted mean. Root of covariance of new information and mutual covariance ; Step 5: State Update: Calculate Kalman Gain Update the state matrix The square root of the covariance ; Step 6: Noise Update: Evaluate whether the state-space equation parameters have stabilized: if so, it indicates that the estimation has reached optimal performance, keeping the process noise constant; If not, it indicates that the state-space equation parameters continue to fluctuate, and the Sage-Husa estimator is used to update the process noise in real time. until the parameters of the state-space equations reach a steady state; Step 7: Repeat steps 3 to 6 for each sample point until all sample points have been processed and stable nonlinear aerodynamic damping parameters are obtained. Step 8: Calculate the nonlinear aerodynamic damping using the obtained stable nonlinear aerodynamic damping parameters; In step one, the relationship between nonlinear aerodynamic damping and structural vibration amplitude is as follows: in: Nonlinear aerodynamic damping; air density; The width of the structure; This represents the effective structural mass per unit height; The amplitude of structural vibration; These are nonlinear aerodynamic damping parameters, and all are constants; The measurement equation is defined as: in: , and Indicates displacement, velocity, and acceleration; The state-space equations are: in: , , , and These are the structural aerodynamic damping parameters; The external load is defined as zero-mean Gaussian white noise; Indicates the modal angular frequency. For frequency; and: ; ; ; ; ; These are nonlinear aerodynamic damping parameters, and all are constants; This indicates the structural damping ratio.

2. The highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter as described in claim 1, characterized in that: In step three, the time update method consists of the following steps: 31) Define the initial state estimate and the initial state covariance matrix definition: in: Represent the initial state space equations; Indicates taking the expected value; Indicates taking the covariance; We obtain the state space vector and the measurement vector: in: and At time step The state vector and measurement vector at the location; and Represents nonlinear state transition functions and nonlinear measurement functions; It is the current time step; and These are the covariance matrices. and Uncorrelated zero-mean Gaussian white noise; For process noise covariance; To measure the noise covariance; 32) Using the radial volume rule of a sphere to generate volume points. : in: yes orthogonal-triangular decomposition, Here is the state covariance matrix; It is a set of volume points; It is the identity matrix The 𝑖 column; 33) Through state transition functions Propagation volume point: in: The time step obtained through propagation via the state transition function is The new volume point; The time step is The volume point at that location; 34) Predicted state mean and the square root of the state covariance : in: in: Represents the Cholesky decomposition function; This represents the QR decomposition of the matrix; for Cholesky decomposition; This is process noise; This is the weight matrix.

3. The highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter as described in claim 1, characterized in that: In step four, the method for measurement and updating is as follows: 41) Using the mean of the state The square root of the state covariance Recalculate the volume points: 42) By measuring the function Propagation volume point: 43) Predict the mean value of measurements from the transformed volume points Root of covariance of new information and mutual covariance : ， ， in: for Cholesky decomposition, For measuring noise; For measuring the weight matrix; This is the new volume point obtained through propagation of the measurement function.

4. The highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter according to claim 1, characterized in that: In step five, the Kalman gain for: State matrix Updated to: Square root of state covariance Updated to: 。 5. The highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filter according to claim 1, characterized in that: In step six, the noise update method is as follows: The noise in the process is updated using the Sage-Husa estimator. : in: in: Forgetting factor; This is an empirical scaling factor; For the new information sequence; The noise in the process is updated iteratively using the Sage-Husa estimator. Achieve the preset After that, using preset The data accumulated during the next iteration is used to evaluate whether the parameters of the state-space equation have stabilized. If in the current number In the next iteration If the ratio of the standard deviation to the mean of the parameter satisfies: in: Indicates the first The standard deviation of the parameters in the next iteration; Indicates the first The average value of the parameters in each iteration; For the current number Next to the front The standard deviation of the parameters in the next iteration; For the current number Next to the front The average value of the parameters in each iteration; The preset threshold; The state-space equation parameters are now stable, and the process noise remains constant. Otherwise, it indicates that the state-space equation parameters are unstable and the process noise is high. Updated to: in: This is an empirical scaling factor that is introduced.

6. A highly flexible nonlinear aerodynamic damping estimation system based on the Sage-Husa algorithm and square root ductile Kalman filtering, characterized in that: It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program and implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root ductile Kalman filtering as described in any one of claims 1-5.

7. A storage medium, characterized in that: The storage medium stores a computer program, which, when executed by a processor, implements the highly flexible nonlinear aerodynamic damping estimation method based on the Sage-Husa algorithm and square root commutative Kalman filtering as described in any one of claims 1-5.

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